Acoustic vibration analysis method for axial pressure orthotropic cylindrical shell under action of turbulent boundary layer

Abstract

The invention discloses a sound and vibration analysis method for an axial-pressure orthotropic cylindrical shell under the action of a turbulent boundary layer, which combines a turbulent boundary layer semi-empirical model with a symplectic space wave method, and solves the random sound and vibration response of the axial-pressure orthotropic cylindrical shell under the action of the turbulent boundary layer by utilizing the symplectic retention property of symplectic space. The method is not only high in calculation precision and calculation speed, but also capable of processing any boundary condition and analyzing the influence of the axial pressure on the sound vibration result.

Description

Technical Field

[0001] The present invention belongs to the technical field of acoustic vibration analysis of cylindrical shell structures in industrial equipment, and in particular relates to an acoustic vibration analysis method for an axially compressed orthotropic cylindrical shell under the action of a turbulent boundary layer. Background Art

[0002] As a typical random excitation, the turbulent boundary layer is one of the main sources of noise during the operation of industrial equipment such as rockets and submarines. Cylindrical shell structures are widely used in industrial equipment manufacturing and other fields due to their simple preparation and excellent mechanical properties. These cylindrical shells usually work in very complex environments, such as the random pressure of the turbulent boundary layer, and are subjected to different dynamic loads. For the acoustic vibration analysis of cylindrical shells under the action of turbulent boundary layers, the existing mainstream analytical methods include the modal superposition method, and the numerical method is mainly the finite element method. In addition, the buckling, free vibration and forced vibration problems based on various thin shell theories have been effectively solved by the symplectic algorithm, and its efficiency and accuracy have been improved compared with the existing methods.

[0003] However, existing technologies still have many deficiencies when dealing with the acoustic vibration problems of cylindrical shells under the action of turbulent boundary layers. The modal superposition method has low calculation accuracy and efficiency due to its modal truncation problem, and can only handle simply supported boundary conditions, which limits its application under complex boundary conditions. The finite element method requires a large amount of meshing when solving high-frequency vibrations, which makes the calculation efficiency extremely low and difficult to meet the needs of fast and efficient analysis in engineering practice. In addition, there are currently few related technologies for the random acoustic vibration problems of cylindrical shells under the action of turbulent boundary layers, and there is a lack of efficient and accurate solutions. Summary of the Invention

[0004] In order to solve the above technical problems, the present invention proposes an acoustic vibration analysis method for an axially compressed orthotropic cylindrical shell under the action of a turbulent boundary layer to solve the problems existing in the above-mentioned prior art.

[0005] To achieve the above objectives, in a first aspect, the present invention provides a method for vibroacoustic analysis of an axially compressed orthotropic cylindrical shell under the action of a turbulent boundary layer, comprising:

[0006] Input the geometric and material parameters of the orthotropic cylindrical shell, various parameters of the turbulent boundary layer, axial pressure parameters, various parameters of the acoustic cavity in the cylindrical shell, and the response positions of vibration and sound radiation. Based on the semi-empirical model of the turbulent boundary layer, the random acoustic and vibroacoustic response of the orthotropic cylindrical shell under the action of the turbulent boundary layer is converted into a steady-state response.

[0007] Based on the Kirchhoff-love classical thin shell theory, the governing equations of orthotropic cylindrical shells in the Lagrange system are established;

[0008] Transforming the governing equations of the orthotropic cylindrical shell under the Lagrange system into the Hamilton system;

[0009] In Hamiltonian system, the simple harmonic response of the orthotropic cylindrical shell is solved by wave propagation analysis in symplectic space.

[0010] Using Kirchhoff-Helmholtz integral, the sound pressure value of the inner acoustic cavity generated by the simple harmonic response of the vibration is obtained;

[0011] Superimposing the steady-state responses, solving and outputting the random vibration and acoustic radiation responses of the orthotropic cylindrical shell;

[0012] By changing the axial pressure value, the influence of the axial pressure on the random vibration and sound radiation response is analyzed.

[0013] Preferably, the geometric parameters include the length, radius and thickness of the cylindrical shell;

[0014] The material parameters include material density, film stiffness and bending stiffness;

[0015] The parameters of the turbulent boundary layer include the autopower spectrum density of the wall pressure, the spatial correlation coefficients in the axial and circumferential directions, and the convection velocity;

[0016] The axial pressure parameter is the axial pressure acting on the cylindrical shell;

[0017] The parameters of the acoustic cavity in the cylindrical shell include density of the fluid medium, acoustic cavity damping and sound velocity;

[0018] The vibration and sound radiation response positions are specific positions on the cylindrical shell for measuring the vibration and sound radiation responses.

[0019] Preferably, according to the semi-empirical model of the turbulent boundary layer, the step of converting the random acoustic vibration response of the orthotropic cylindrical shell under the action of the turbulent boundary layer into a steady-state response includes:

[0020] The arbitrary response of the cylindrical shell structure is:

[0021]

[0022] where S and s represent the position (x, θ), h(S, s, t-τ) is the unit impulse response at point S at time t caused by the unit impulse force acting at point s at time τ, and Γ is the surface of the structure.

[0023] q(S, t) is a random response function in time and space. The cross-correlation function of the structure response at point S1 and point S2 is written as

[0024]

[0025] Where E[] is the expectation operator, E[p(s1,τ1)p(s2,τ2)] is the cross-correlation function of pressure p(s,τ);

[0026] Through the Wiener-Khinchin relationship, we get:

[0027]

[0028] Among them, τ=τ2-τ1 and ξ=s2-s1,S pp (ξ, ω) is the cross-power spectral density of the turbulent boundary layer;

[0029] Further we get:

[0030]

[0031] The formula of the semi-empirical model is:

[0032]

[0033] Among them, Φ pp (ω) is the auto-power spectrum density of the wall pressure; c x and c θ are the spatial correlation coefficients of the wall pressure in the axial and circumferential directions, respectively; ξ x and ξ θ Represents the position distance; U c is the convection velocity;

[0034] Further we get:

[0035]

[0036] According to the Wiener-Khinchin relationship, the cross power spectrum density of the response q(s,t) is

[0037]

[0038] According to the Kirchhoff-Helmholtz integral expression, the sound pressure p radiated by the cylindrical shell vibration to a point r in the internal sound cavity at time t is in (r, t) is:

[0039]

[0040] Where ρ0 is the density of the fluid medium, is the radial acceleration of the cylindrical shell, g(r; S; t-τ) is the Green's function in the time domain, which represents the sound pressure at point r inside the cylindrical shell caused by a source of unit source intensity starting at time τ at point S on the cylindrical shell;

[0041] According to the definition of the cross-correlation function, the sound pressure p at any two points in the internal sound cavity at different times in (r1, t1) and p in The cross-correlation function of (r2, t2) can be expressed as:

[0042]

[0043] in, is the cross-correlation function of the radial vibration acceleration of the cylindrical shell;

[0044] According to the Wiener-Khinchin relationship, the sound pressure p in The cross power spectrum density of (r, t) can be expressed as:

[0045]

[0046] Among them, S qq (S1, S2, ω) is the displacement response cross-power spectrum density obtained in the previous section, and G(r, S, ω) is the Green function in the frequency domain:

[0047]

[0048] Where v is the acoustic cavity damping, k0=ω / c0, k l =ω l / c0, c0 is the speed of sound in the fluid, Λ l =1 / (πR 2 L)∫∫∫ V φ l (r)φ l (r)dV is the normalization factor of the acoustic cavity mode;

[0049] is the l-th order natural frequency of the acoustic cavity;

[0050] φ l (r)=φ jpq (x,θ,r)=cos(jθ)J j (λ jp r)cos[(qπ / L)x] is the lth-order acoustic mode function of the cylindrical acoustic cavity;

[0051] Where j, p and q represent the circumferential wave number, radial quarter wave number and axial half wave number of the acoustic cavity respectively; J j (λ jp r) indicates that the quantity is λ jp The j-th order Bessel function of r, λ jp J′ j (λ jp R) = the pth root of 0;

[0052] Still considering the semi-empirical model proposed by Corcos, we can further obtain:

[0053]

[0054] The sound pressure is solved using the Green function and the simple harmonic response function obtained in the previous section; further solution is used to obtain the random sound pressure response of the structure.

[0055] Preferably, the cylindrical shell control equations include geometric equations, internal force expressions and equilibrium equations.

[0056] Preferably, the step of converting the governing equations of the orthotropic cylindrical shell under the Lagrange system into the Hamilton system comprises:

[0057] Take the state vector as:

[0058]

[0059] Transforming the governing equation into the Hamiltonian system, we can obtain:

[0060]

[0061] Where H is the Hamilton operator matrix, f={0 0 0 0 0 0 p MN 0} T is the external excitation vector;

[0062] Using the separation of variables method, the state vector can be written as:

[0063]

[0064] where μ n represents the wave propagation parameter in the axial direction of the cylindrical shell;

[0065] According to the superposition principle of solutions of linear differential equations, the solution corresponding to each circumferential wave number n satisfies the equation, and we get:

[0066]

[0067] Further, we get:

[0068]

[0069] Further, we get:

[0070] Hη n =μη n

[0071] According to the periodic boundary conditions of the cylindrical shell in the circumferential direction, η n Expressed as:

[0072]

[0073] in is an imaginary unit;

[0074] Further, we get

[0075]

[0076] in, is the Hamilton matrix, which is only related to the material properties of the cylindrical shell, the circumferential wave number n and the excitation frequency, μ n is the wave propagation parameter of the forward wave, φ n Arranged in order.

[0077] Preferably, the step of solving the vibration harmonic response of the orthotropic cylindrical shell using wave propagation analysis in symplectic space includes:

[0078] After normalizing the waveform, the sorted waveform matrix satisfies

[0079]

[0080] in

[0081]

[0082] J8 is the unit symplectic matrix, satisfying -J=J -1 =J T ;

[0083] The expansion of the state vector under the orthogonal wave basis is:

[0084]

[0085] where a n is the generalized coordinate vector of z;

[0086] Expanding the excitation force under the orthogonal wave basis, we get:

[0087]

[0088] According to the conjugate symplectic orthogonal relation, multiply both ends by And integrating from 0 to 2π along the circumferential direction, we get:

[0089]

[0090] Due to the spatial harmonic pressure p MN It is exponentially distributed in the circumferential direction, like e -iNθ , again using the conjugate symplectic orthogonal relation, we get:

[0091]

[0092] Among them, Λ n is a diagonal matrix. According to the solution method of the first-order linear non-homogeneous differential equation, it can be written as the sum of the homogeneous solution and the non-homogeneous solution. Since the spatial harmonic pressure p MN The distribution is exponential in the axial direction, as The solution is of the form:

[0093] a n =B n A n -(iα M I8+Λ n ) -1 b n

[0094] in And A n is the unknown coefficient vector determined by the boundary conditions;

[0095] There are four displacement constraints and four force constraints on the cross section of the orthotropic cylindrical shell. The combination of the eight constraints represents any boundary condition, and any displacement constraint and the corresponding force constraint cannot exist at the same time.

[0096] The boundary conditions can be expressed as:

[0097] γz(x,θ)=γφ n e -inθ a n (x) = 0 8×1

[0098] Where γ is the indicator matrix of the boundary conditions;

[0099] Using the conjugate symplectic orthogonal relation, multiply both ends by And integrating from 0 to 2π along the circumferential direction, we get:

[0100]

[0101] Wherein, the subscripts L and R represent the left and right ends of the cylindrical shell respectively; further find A n , thus obtaining the generalized coordinate vector a n ;

[0102] According to a n The state vector z is obtained, the displacement response of the orthotropic cylindrical shell is obtained, and the internal force response is obtained at the same time.

[0103] Preferably, the Kirchhoff-Helmholtz integral is used to solve the formula for the inner acoustic cavity sound pressure value generated by the simple harmonic response:

[0104] pMN =-ρ0ω 2 ∫ Γ G MN (S,ω)G(r,S,ω)dS

[0105] Among them, G MN (r, S, ω) is the Green function in the frequency domain.

[0106] In a second aspect, the present invention further discloses a computer device comprising a memory, a processor, and a computer program stored in the memory, wherein the processor executes the computer program to implement the steps of the method described in the first aspect.

[0107] In a third aspect, the present invention further discloses a computer-readable storage medium having a computer program stored thereon, which implements the steps of the method described in the first aspect when executed by a processor.

[0108] In a fourth aspect, the present invention further discloses a computer program product, comprising a computer program, which implements the steps of the method described in the first aspect when executed by a processor.

[0109] Compared with the prior art, the present invention has the following advantages and technical effects:

[0110] The present invention provides an acoustic-vibration analysis method for an orthotropic cylindrical shell under axial compression under a turbulent boundary layer, comprising the following steps: first, inputting geometric parameters and material parameters of the orthotropic cylindrical shell, various parameters of the turbulent boundary layer, axial pressure parameters, various parameters of the acoustic cavity in the cylindrical shell, and response positions of vibration and sound radiation; second, converting the random acoustic-vibration response of the orthotropic cylindrical shell under the turbulent boundary layer into a steady-state response based on a semi-empirical model of the turbulent boundary layer; and then, establishing the governing equations of the orthotropic cylindrical shell under the Lagrange system based on the Kirchhoff-Love classical thin shell theory. Furthermore, the control equations of the orthotropic cylindrical shell under the Lagrange system are transformed into the Hamilton system; again, under the Hamilton system, the wave propagation analysis in the symplectic space is used to solve the vibration harmonic response of the orthotropic cylindrical shell; the Kirchhoff-Helmholtz integral is used to solve the sound pressure value of the inner acoustic cavity generated by the vibration harmonic response; finally, the steady-state response is superimposed to solve and output the random vibration and sound radiation response of the orthotropic cylindrical shell; by changing the axial pressure value, the influence of the axial pressure on the random vibration and sound radiation response is analyzed.

[0111] Based on the symplectic algorithm, this paper proposes an innovative approach that combines a semi-empirical model of the turbulent boundary layer with the symplectic space wave method. Leveraging the symplectic-preserving properties of symplectic space, this paper develops a method for solving the random vibroacoustic response of an orthotropic cylindrical shell subjected to axial compression in a turbulent boundary layer. This method not only offers high accuracy and high computation speed, but also handles arbitrary boundary conditions and simultaneously analyzes the impact of axial compression on the vibroacoustic response. BRIEF DESCRIPTION OF THE DRAWINGS

[0112] The accompanying drawings, which constitute part of this application, are intended to provide a further understanding of this application. The exemplary embodiments and descriptions of this application are intended to explain this application and do not constitute an improper limitation on this application. In the accompanying drawings:

[0113] Figure 1 Schematic diagram of an orthotropic cylindrical shell and coordinates under the action of a turbulent boundary layer according to an embodiment of the present invention;

[0114] Figure 2 A schematic diagram of the positional relationship between excitation and response according to an embodiment of the present invention;

[0115] Figure 3 Schematic diagram of the displacement autopower spectrum density at (0.3L, 0.4π) obtained by the embodiment of the present invention and the modal superposition method;

[0116] Figure 4 The solution obtained by the embodiment of the present invention and the modal superposition method is Schematic diagram of the sound pressure level of the sound pressure auto-power spectrum density at ;

[0117] Figure 5 Schematic diagram of random response under different axial pressures under simply supported boundary conditions at both ends of the embodiment of the present invention, (a) is a schematic diagram of the displacement power spectrum density at (0.3L, 0.4π) under different axial pressures, (b) is a schematic diagram of the displacement power spectrum density at different axial pressures Schematic diagram of the sound pressure level and the sound pressure power spectrum density at . DETAILED DESCRIPTION

[0118] It should be noted that, in the absence of conflict, the embodiments and features of the embodiments in this application can be combined with each other. The present application will be described in detail below with reference to the accompanying drawings and in combination with the embodiments.

[0119] It should be noted that the steps shown in the flowcharts of the accompanying drawings can be executed in a computer system such as a set of computer-executable instructions, and that, although a logical order is shown in the flowcharts, in some cases, the steps shown or described can be executed in an order different from that shown here.

[0120] Example 1

[0121] like Figure 1-2As shown, this embodiment provides an acoustic vibration analysis method for an axially compressed orthotropic cylindrical shell under the action of a turbulent boundary layer, including:

[0122] S1. Input the geometric and material parameters of the orthotropic cylindrical shell, various parameters of the turbulent boundary layer, axial pressure parameters, various parameters of the acoustic cavity in the cylindrical shell, and the response positions of vibration and sound radiation. Based on the semi-empirical model of the turbulent boundary layer, the random response of the orthotropic cylindrical shell under the action of the turbulent boundary layer is converted into a superposition of steady-state responses.

[0123] Considering the random pressure field p(s, τ) acting on the turbulent boundary layer, Figure 1 Consider a cylindrical shell with an axial pressure N0 acting per unit length in the x-direction. The shell has a length of L, a radius of R, and a thickness of h. s is the location of the external excitation, and t is the time. The arbitrary response of the structure can be expressed as the following convolution integral.

[0124]

[0125] Where S and s represent the position (x, θ), h(S, s, t-τ) is the unit impulse response at point S at time t caused by the unit impulse force acting at point s at time τ, and Γ is the surface of the structure. And h(S, s, t-τ) and p(s, τ) satisfy the following relationship

[0126] p(s,τ)=0τ<0

[0127] h(S,s,t-τ)=0t<τ (2)

[0128] The above relationship indicates that the response always appears after the stimulus. Using this relationship, we can get

[0129]

[0130] Since q(S, t) is a random response function in time and space, the cross-correlation function of the structure response at point S1 and point S2 is written as

[0131]

[0132] Where E[ ] is the expectation operator. Therefore, E[p(s1, τ1)p(s2, τ2)] is the cross-correlation function of pressure p(s, τ), which is denoted by R pp (S1, S2; t1, t2). Through the Wiener-Khinchin relationship, we can get;

[0133]

[0134] Among them, τ=τ2-τ1 and ξ=s2-s1,S pp(ξ, ω) is the cross power spectral density of the turbulent boundary layer.

[0135] Substituting (5) into (4), we can obtain;

[0136]

[0137] in,() * is a complex conjugate relationship. And H(S, s, ω) is the frequency response function, which satisfies the unit impulse response function;

[0138]

[0139] This embodiment adopts the semi-empirical model proposed by Corcos, which is as follows

[0140]

[0141] Among them, Φ pp (ω) is the auto-power spectrum density of the wall pressure; c x and c θ are the spatial correlation coefficients of the wall pressure in the axial and circumferential directions, respectively; ξ x and ξ θ Represents the position distance, denoted as ξ x =x2-x1 and ξ θ =θ2-θ1; U c is the convection velocity. This semi-empirical model can be expanded into the Fourier series form in the axial and circumferential directions as follows

[0142]

[0143] Among them, ξ x and ξ θ The ranges are -L to L and -π to π respectively, so S ppx (M) and S ppθ (N) can be expressed by finite integral as

[0144]

[0145] in

[0146]

[0147]

[0148] in

[0149]

[0150] Substituting (9) into (6) yields

[0151]

[0152] in

[0153]

[0154] is the simple harmonic pressure in space and time The simple harmonic response function under the action of.

[0155] According to the Wiener-Khinchin relationship, the cross power spectrum density of the response q(S, t) is

[0156]

[0157] When S=S1=S2, the above formula is the autopower spectrum density.

[0158] Therefore, the solution of the random displacement response under the action of the turbulent boundary layer will be converted into the solution of the simple harmonic response by expanding the displacement response autopower spectral density into a Fourier series.

[0159] According to the Kirchhoff-Helmholtz integral expression, the sound pressure p radiated by the cylindrical shell vibration to a point r in the internal sound cavity at time t is in (r, t) is

[0160]

[0161] Where ρ0 is the density of the fluid medium, is the radial acceleration of the cylindrical shell, g(r, S; t-τ) is the Green function in the time domain, which represents the sound pressure generated by a source with unit source intensity at point S on the cylindrical shell starting at time τ at the internal r. The position relationship is as follows Figure 2 .

[0162] According to the definition of the cross-correlation function, the sound pressure p at any two points in the internal sound cavity at different times in (r1, t1) and p in The cross-correlation function of (r2, t2) can be expressed as

[0163]

[0164] in, is the cross-correlation function of the radial vibration acceleration of the cylindrical shell.

[0165] According to the Wiener-Khinchin relationship, the sound pressure p in The cross power spectral density of (r, t) can be expressed as

[0166]

[0167] Among them, S qq(S1, S2, ω) is the displacement response cross power spectrum density obtained in the previous section, and G(r, S, ω) is the Green function in the frequency domain

[0168]

[0169] Where υ is the acoustic cavity damping, k0=ω / c0, k l =ω l / c0, c0 is the speed of sound in the fluid, Λ l =1 / (πR 2 L)∫∫∫ V φ l (r)φ l (r)dV is the normalization factor of the acoustic cavity mode.

[0170]

[0171] is the lth-order natural frequency of the acoustic cavity.

[0172] φ l (r)=φ jpq (x,θ,r)=cos(jθ)J j (λ jp r)cos[(qπ / L)x] (22)

[0173] is the l-th order acoustic mode function of the cylindrical acoustic cavity. j, p and q represent the circumferential wave number, radial quarter wave number and axial half wave number of the acoustic cavity respectively; J j (λ jp r) indicates that the quantity is λ jp The j-th order Bessel function of r, λ jp J′ j (λ jp R)=0's pth root.

[0174] Still considering the semi-empirical model proposed by Corcos, substituting (16) and (20) into (19) we can obtain

[0175]

[0176] in

[0177] p MN (r,ω)=-ρ0ω 2 ∫ Γ G MN (S;ω)G(r;S;ω)dS (24)

[0178] When r=r1=r2, equation (23) is the autopower spectrum density.

[0179] From Equation (23), we can see that the sound pressure can be solved by Green’s function and the simple harmonic response function obtained in the previous section. Then, we can solve Equation (23) to obtain the random sound pressure response of the structure.

[0180] S2. Based on the Kirchhoff-Love classical thin shell theory, the governing equations of orthotropic cylindrical shells in the Lagrange system are established;

[0181] The geometric equation is as follows:

[0182]

[0183]

[0184] Where u, υ and w are the displacements along the three directions of the coordinate system.

[0185] Internal force expression:

[0186]

[0187]

[0188] Among them A ij ,D ij (i, j = 1, 2, 6) are the film stiffness and bending stiffness.

[0189] Balanced equation:

[0190] The vibration equilibrium equation of an axially compressed orthotropic cylindrical shell is:

[0191]

[0192] Where ρ is the material density of the cylindrical shell, N0 is the axial pressure, and p MN is the external excitation force caused by the turbulent boundary layer.

[0193] S3. Transform the governing equations of orthotropic cylindrical shells under the Lagrange system into the Hamilton system.

[0194] Take the state vector

[0195]

[0196] Transforming the governing equation into the Hamiltonian system, we can obtain

[0197]

[0198] Where H is the Hamilton operator matrix. f={0 0 0 0 0 0 p MN 0} T is the external excitation vector.

[0199] Considering the homogeneous form of Equation (31), the state vector can be written as

[0200]

[0201] where μ n represents the wave propagation parameter in the axial direction of the cylindrical shell.

[0202] According to the superposition principle of solutions of linear differential equations, the solution corresponding to each circumferential wave number n can satisfy the equation, so we get

[0203]

[0204] Substituting (33) into (32), we can obtain

[0205]

[0206] Substituting (34) and (33) into the homogeneous form of (31), we can obtain

[0207] Hη n =μη n (35)

[0208] According to the periodic boundary conditions of the cylindrical shell in the circumferential direction, η n It can be expressed as

[0209]

[0210] in Is an imaginary unit.

[0211] Substituting (36) into (35), we can obtain

[0212]

[0213] in, is the Hamiltonian matrix, which is only related to the material properties of the cylindrical shell, the circumferential wave number n and the excitation frequency. It has a special property that its eigenvalues always appear in pairs, namely the wave propagation parameter μ of the forward wave n and the wave propagation parameter of the reverse wave - μ n , can be arranged in order as A n Its eigenvectors are waveforms, arranged in the same order as Φ n .

[0214] S4. In the Hamiltonian system, the wave propagation analysis in symplectic space is used to solve the simple harmonic vibration response of the orthotropic cylindrical shell.

[0215] After normalizing the waveform, the sorted waveform matrix satisfies

[0216]

[0217] in

[0218]

[0219] J8 is the unit symplectic matrix, satisfying -J=J -1 =J T .

[0220] Substituting (33) and (36) into (34), we can obtain the expansion of the state vector in the orthogonal wave basis:

[0221]

[0222] where a n is the generalized coordinate vector of z.

[0223] Expand the excitation force under the orthogonal wave basis and we can get

[0224]

[0225] in

[0226] Λ n =diag[μ n,1 μ n,2 μ n,3 μ n,4 -μ n,1 -μ n,2 -μ n,3 -μ n,4 ] (42)

[0227]

[0228] According to the conjugate symplectic orthogonal relation, we multiply both ends of equation (41) by And integrating from 0 to 2π along the circumferential direction, we can get

[0229]

[0230] Due to the spatial harmonic pressure p MN It is exponentially distributed in the circumferential direction, like e -iNθ According to formula (44), it is obvious that only when n=N, b n is a non-zero vector. This means that both Equations (40) and (41) will not be truncated, which greatly reduces the amount of calculation.

[0231] Substituting equations (40) and (41) into equation (31), and again using the conjugate symplectic orthogonal relation, we can obtain:

[0232]

[0233] Among them, Λ n is the diagonal matrix in Equation (42). Therefore, Equation (45) is 8 decoupled non-homogeneous differential equations. According to the solution method of first-order linear non-homogeneous differential equations, Equation (45) can be written as the sum of homogeneous solutions and non-homogeneous solutions. MN The distribution is exponential in the axial direction, as The solution is of the form

[0234] a n =B n A n -(iα M I8+Λ n ) -1 b n (46)

[0235] in And A n is the unknown coefficient vector determined by the boundary conditions.

[0236] The cross section of an orthotropic cylindrical shell has four displacement constraints and four force constraints. The combination of these eight constraints can represent any boundary condition, and any displacement constraint and the corresponding force constraint cannot coexist.

[0237] The boundary conditions can be expressed as

[0238] γz(x,θ)=γΦ n e -inθ a n (x) = 0 8×1 (47)

[0239] Where γ is the indicator matrix of the boundary conditions;

[0240] For simply supported boundary conditions, γ = diag[0 1 1 0 1 0 0 1];

[0241] For the clamped boundary condition, γ=diag[1 1 1 1 0 0 0 0];

[0242] For free boundary conditions, γ = diag[0 0 0 0 1 1 1 1].

[0243] Using the conjugate symplectic orthogonal relation, we multiply both ends of Equation (47) by And integrating from 0 to 2π along the circumferential direction, we get:

[0244]

[0245] Wherein, the subscripts L and R represent the left and right ends of the cylindrical shell, respectively. Substituting equation (46) into equation (48), we can obtain A n , thus obtaining the generalized coordinate vector a n

[0246] will a n Substituting into Equation (42), the state vector z can be obtained. From Equation (42), it can be seen that this method can not only obtain the displacement response of the orthotropic cylindrical shell, but also obtain the internal force response at the same time.

[0247] S5. Using the Kirchhoff-Helmholtz integral, the sound pressure value of the inner acoustic cavity generated by the simple harmonic response is obtained as follows:

[0248] p MN =-ρ0ω 2 ∫ Γ G MN (S, ω)G(r, S, ω)dS (49)

[0249] Among them, G MN (r, S, ω) is the Green function in the frequency domain.

[0250] S6. Superimpose the steady-state responses, solve and output the random vibration and acoustic radiation responses of the orthotropic cylindrical shell.

[0251] S7. By changing the axial pressure value, the Hamilton operator matrix can be changed, and the influence of the axial pressure on the random response can be analyzed at the same time.

[0252] The terms of the Hamiltonian operator matrix for an axially compressed orthotropic cylindrical shell are:

[0253]

[0254]

[0255] This embodiment not only has high computational efficiency, but also provides analytical results. It can also handle arbitrary boundary conditions and analyze the effect of axial pressure on random acoustic vibration response.

[0256] This embodiment can handle arbitrary boundary conditions, including classical boundary conditions and non-classical elastic boundary conditions.

[0257] The displacement power spectrum density calculated by the method of this embodiment at (0.3L, 0.4π) is compared with the modal superposition method under different cutoffs, such as Figure 3 It can be seen that as the number of modes increases, the results of the modal superposition method converge to the method of this embodiment, and the higher the excitation frequency, the more modes are required to obtain converged results. Figure 4 Shown in The sound pressure level of the sound pressure auto-power spectrum density calculated at Figure 3 The phenomenon is the same.

[0258] Similarly, for the convenience of displaying the results, the sound pressure level of the sound pressure autopower spectrum density is defined as (in dB)

[0259]

[0260] Tables 1 and 2 show the time obtained by calculating the simple harmonic response using this method and the modal superposition method under different truncation conditions. The simple harmonic response is calculated in the range of 1-1000Hz with a step size of 1Hz. Obviously, the time of the modal superposition method increases linearly with the increase in the number of modes, but the method of this embodiment will always maintain the same time because there is no truncation. Therefore, the method of this embodiment has higher computational efficiency than the modal superposition method, especially for structural vibration analysis under broadband excitation, such as turbulent boundary layer excitation. After convergence analysis, m max =32, the modal superposition method reaches convergence.

[0261] Table 1

[0262] M=1,N=2

[0263]

[0264] Table 2

[0265] M=4, N=4

[0266]

[0267]

[0268] Investigate the random response of a cylindrical shell to axial pressure under a turbulent boundary layer. Use different boundary conditions and the maximum axial pressure value is expressed as N cr =2×10 7 N / m. Figure 5 (a) and Figure 5 (b) The power spectrum density S of the random displacement response to axial pressure at (0.3L, 0.4π) under different boundary conditions is given. qq (S1, S2, ω) and The influence of the sound pressure level SPL on the sound pressure self-power spectrum density at S. It can be seen that the change of axial pressure has a great influence on the qq (S1, S2, ω) has a great influence on SPL. As the axial pressure increases, S qq The peak values of (S1, S2, ω) and SPL shift to the left.

[0269] Example 2

[0270] This embodiment further discloses a computer device, including a memory, a processor, and a computer program stored in the memory, wherein the processor executes the computer program to implement the steps of the method described in the first embodiment.

[0271] Example 3

[0272] This embodiment further discloses a computer-readable storage medium on which a computer program is stored. When the computer program is executed by a processor, the steps of the method described in the first embodiment are implemented.

[0273] Example 4

[0274] This embodiment further discloses a computer program product, including a computer program, which implements the steps of the method described in the first embodiment when executed by a processor.

[0275] The above are merely preferred embodiments of the present application, but the scope of protection of the present application is not limited thereto. Any changes or substitutions that can be easily conceived by a person skilled in the art within the technical scope disclosed in this application should be included in the scope of protection of the present application. Therefore, the scope of protection of the present application should be based on the scope of protection of the claims.

Claims

1. A method for vibroacoustic analysis of an axially compressed orthotropic cylindrical shell under turbulent boundary layer, characterized in that: The following steps are involved: Input the geometric and material parameters of the orthotropic cylindrical shell, various parameters of the turbulent boundary layer, axial pressure parameters, various parameters of the acoustic cavity in the cylindrical shell, and the response positions of vibration and sound radiation. Based on the semi-empirical model of the turbulent boundary layer, the random acoustic and vibroacoustic response of the orthotropic cylindrical shell under the action of the turbulent boundary layer is converted into a steady-state response. Based on the Kirchhoff-love classical thin shell theory, the governing equations of orthotropic cylindrical shells in the Lagrange system are established; Transforming the governing equations of the orthotropic cylindrical shell under the Lagrange system into the Hamilton system; In Hamiltonian system, the simple harmonic response of the orthotropic cylindrical shell is solved by wave propagation analysis in symplectic space. Using Kirchhoff-Helmholtz integral, the sound pressure value of the inner acoustic cavity generated by the simple harmonic response of the vibration is obtained; Superimposing the steady-state responses, solving and outputting the random vibration and acoustic radiation responses of the orthotropic cylindrical shell; By changing the axial pressure value, the influence of the axial pressure on the random vibration and sound radiation response is analyzed.

2. The method according to claim 1, characterized in that The geometric parameters include the length, radius and thickness of the cylindrical shell; The material parameters include material density, film stiffness and bending stiffness; The parameters of the turbulent boundary layer include the autopower spectrum density of the wall pressure, the spatial correlation coefficients in the axial and circumferential directions, and the convection velocity; The axial pressure parameter is the axial pressure acting on the cylindrical shell; The parameters of the acoustic cavity in the cylindrical shell include density of the fluid medium, acoustic cavity damping and sound velocity; The vibration and sound radiation response positions are specific positions on the cylindrical shell for measuring the vibration and sound radiation responses.

3. The method according to claim 1, characterized in that Based on the semi-empirical model of the turbulent boundary layer, the steps for converting the random acoustic vibration response of an orthotropic cylindrical shell under the action of a turbulent boundary layer into a steady-state response include: The arbitrary response of the cylindrical shell structure is: where S and s represent the position (x, θ), h(S, s, t-τ) is the unit impulse response at point S at time t caused by the unit impulse force acting at point s at time τ, and Γ is the surface of the structure. q(S, t) is a random response function in time and space. The cross-correlation function of the structure response at point S1 and point S2 is written as Where E[] is the expectation operator, E[p(s1,τ1)p(s2,τ2)] is the cross-correlation function of pressure p(s,τ); Through the Wiener-Khinchin relationship, we get: Among them, τ=τ2-τ1 and ξ=s2-s1,S pp (ξ, ω) is the cross-power spectral density of the turbulent boundary layer; Further we get: The formula of the semi-empirical model is: Among them, Φ pp (ω) is the auto-power spectrum density of the wall pressure; c x and c θ are the spatial correlation coefficients of the wall pressure in the axial and circumferential directions, respectively; ξ x and ξ θ Represents the position distance; U c is the convection velocity; Further we get: According to the Wiener-Khinchin relationship, the cross power spectrum density of the response q(S, t) is According to the Kirchhoff-Helmholtz integral expression, the sound pressure p radiated by the cylindrical shell vibration to a point r in the internal sound cavity at time t is in (r, t) is: Where ρ0 is the density of the fluid medium, is the radial acceleration of the cylindrical shell, g(r; S; t-τ) is the Green's function in the time domain, which represents the sound pressure at point r inside the cylindrical shell caused by a source of unit source intensity starting at time τ at point S on the cylindrical shell; According to the definition of the cross-correlation function, the sound pressure p at any two points in the internal sound cavity at different times in (r1, t1) and p in The cross-correlation function of (r2, t2) can be expressed as: in, is the cross-correlation function of the radial vibration acceleration of the cylindrical shell; According to the Wiener-Khinchin relationship, the sound pressure p in The cross power spectrum density of (r, t) can be expressed as: Among them, S qq (S1, S2, ω) is the displacement response cross-power spectrum density obtained in the previous section, and G(r, S, ω) is the Green function in the frequency domain: Where v is the acoustic cavity damping, k0=ω / c0, k l =ω l / c0, c0 is the speed of sound in the fluid, Λ l =1 / (πR 2 L)∫∫∫ V φ l (r)φ l (r)dV is the normalization factor of the acoustic cavity mode; is the l-th order natural frequency of the acoustic cavity; φ l (r)=φ jpq (x,θ,r)=cos(jθ)J j (λ jp r)cos[(qπ / L)x] is the lth-order acoustic mode function of the cylindrical acoustic cavity; Where j, p and q represent the circumferential wave number, radial quarter wave number and axial half wave number of the acoustic cavity respectively; J j (λ jp r) indicates that the quantity is λ jp r's jth order Bessel function, λ jp J′ j (λ jp R) = the pth root of 0; Still considering the semi-empirical model proposed by Corcos, we can further obtain: The sound pressure is solved using the Green function and the simple harmonic response function obtained in the previous section; further solution is used to obtain the random sound pressure response of the structure.

4. The method according to claim 1, wherein The cylindrical shell control equations include geometric equations, internal force expressions and equilibrium equations.

5. The method according to claim 1, wherein The steps to transform the governing equations of the orthotropic cylindrical shell in the Lagrange system into the Hamilton system include: Take the state vector as: Transforming the governing equation into the Hamiltonian system, we can obtain: Where H is the Hamilton operator matrix, f={0 0 0 0 0 0 p MN 0] T is the external excitation vector; Using the separation of variables method, the state vector can be written as: where μ n represents the wave propagation parameter in the axial direction of the cylindrical shell; According to the superposition principle of solutions of linear differential equations, the solution corresponding to each circumferential wave number n satisfies the equation, and we get: Further, we get: Further, we get: The n =not n According to the periodic boundary conditions of the cylindrical shell in the circumferential direction, η n Expressed as: in is an imaginary unit; Further, we get in, is the Hamilton matrix, which is only related to the material properties of the cylindrical shell, the circumferential wave number n and the excitation frequency, μ n is the wave propagation parameter of the forward wave, Φ n Arranged in order.

6. The method according to claim 1, characterized in that The steps for solving the simple harmonic vibration response of an orthotropic cylindrical shell using wave propagation analysis in symplectic space include: After normalizing the waveform, the sorted waveform matrix satisfies in J8 is the unit symplectic matrix, satisfying -J=J -1 =J T ; The expansion of the state vector under the orthogonal wave basis is: where a n is the generalized coordinate vector of z; Expanding the excitation force under the orthogonal wave basis, we get: According to the conjugate symplectic orthogonal relation, multiply both ends by And integrating from 0 to 2π along the circumferential direction, we get: Due to the spatial harmonic pressure p MN It is exponentially distributed in the circumferential direction, like e -iNθ , again using the conjugate symplectic orthogonal relation, we get: Among them, Λ n is a diagonal matrix. According to the solution method of the first-order linear non-homogeneous differential equation, it can be written as the sum of the homogeneous solution and the non-homogeneous solution. Since the spatial harmonic pressure p MN The distribution is exponential in the axial direction, as The solution is of the form: a n =B n A n -(iα M I8+L n ) -1 b n in And A n is the unknown coefficient vector determined by the boundary conditions; There are four displacement constraints and four force constraints on the cross section of the orthotropic cylindrical shell. The combination of the eight constraints represents any boundary condition, and any displacement constraint and the corresponding force constraint cannot exist at the same time. The boundary conditions can be expressed as: γz(x,θ)=γΦ n e -inθ a n (x)=0 8×1 Where γ is the indicator matrix of the boundary conditions; Using the conjugate symplectic orthogonal relation, multiply both ends by And integrating from 0 to 2π along the circumferential direction, we get: Wherein, the subscripts L and R represent the left and right ends of the cylindrical shell respectively; further find A n , thus obtaining the generalized coordinate vector a n ; According to a n The state vector z is obtained, the displacement response of the orthotropic cylindrical shell is obtained, and the internal force response is obtained at the same time.

7. The method according to claim 1, characterized in that Using the Kirchhoff-Helmholtz integral, the formula for the sound pressure value of the inner acoustic cavity generated by the simple harmonic response is: p MN =-ρ0ω 2 ∫ Γ G MN (S,ω)G(r,S,ω)dS Among them, G MN (r, S, ω) is the Green function in the frequency domain.

8. A computer device comprising a memory, a processor, and a computer program stored in the memory, wherein: The processor executes the computer program to implement the steps of the method according to any one of claims 1 to 7.

9. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by a processor, the steps of the method according to any one of claims 1 to 7 are implemented.

10. A computer program product comprising a computer program, characterized in that When the computer program is executed by a processor, the steps of the method according to any one of claims 1 to 7 are implemented.

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